Shape & Growing Patterns

Matchstick patterns turn shapes into sequences. One square needs 4 sticks — but each NEW square shares a side, so it only adds 3. Sticks: 4, 7, 10, 13… To jump straight to step 10, don't draw: the first square gives 4, then 9 more squares at 3 sticks each — 4 + 27 = 31. Understanding WHY it grows by 3 (the shared side!) is what lets you leap ahead.

A table makes growing patterns obey. List step against count and hunt for the ACROSS rule: how do you turn the step number straight into the count? Here sticks = 3 × step + 1 (check: 3×2+1 = 7 ✓). Now any step is instant: step 10 → 31, step 100 → 301. Down-the-table rules (+3 each time) are slow; across-the-table rules teleport.

Border problems are growing patterns in disguise: a 5×3 pool surrounded by a ring of tiles. Slow way: count tiles one by one (and miscount a corner). Fast way: the whole rectangle including the border is 7×5 = 35 tiles; the pool is 5×3 = 15; border = 35 − 15 = 20. Outer rectangle minus inner rectangle — corners handled automatically, every time.

OC fuses patterns with geometry: join unit squares in a row and watch the perimeter — 4, 6, 8, 10… Why +2 each time? Each join hides TWO sides (one from each square), so a new square brings 4 sides but nets only 2. Across rule: P = 2 × squares + 2. This 'what does joining hide?' reasoning also solves the wire-rectangle questions — and it beats counting sides every time.

The complete OC pattern method: read the first three steps carefully and WRITE the counts; find the growth and understand why (shared sides! hidden joins!); convert to an across-rule and verify it twice; then answer exactly what was asked — which is usually one step beyond the pattern itself. Understanding the WHY is your proof that the rule won't betray you at step 50. Quiz and exercises await!